As thoroughly discussed, e.g., in [1]},
the above spaces satisfy fundamental exact sequence properties.
Introduce an edge approximation operator \(\operatorname{\mathcal {I}^{\mathcal {E}_{h}}}:H^{}(\operatorname{\mathbf {curl}},\Omega )\rightarrow \mathbf {V}_{h}^{\operatorname{edge}}\)  as follows.
The edge approximant \(\operatorname{\mathcal {I}^{\mathcal {E}_{h}}}(\mathbf {E}) := \mathbf {E}_{I\!I}\in \mathbf {V}_{h}^{\operatorname{edge}}\)  of a given vector field \(\mathbf {E}\in H^{}(\operatorname{\mathbf {curl}},\Omega )\)  is defined as the solution to the variational problem
\(\big [\operatorname{\mathbf {curl}}\mathbf {E}_{I\!I},\operatorname{\mathbf {curl}}\mathbf {F}_{h}\big ]_{\operatorname{face}} &= \big (\operatorname{\mathbf {curl}}\mathbf {E},\operatorname{\mathbf {curl}}\mathbf {F}_{h}\big ) && \hspace{-35.56593pt}\forall \mathbf {F}_{h}\in \mathbf {V}_{h}^{\operatorname{edge}}, \\[0.5em]\big [\mathbf {E}_{I\!I},\nabla r_{h}\big ]_{\operatorname{edge}} &= \big (\mathbf {E},\nabla r_{h}\big ) && \hspace{-35.56593pt}\forall r_{h}\in V_{h}^{\operatorname{node}}, \) 
